quality control for differential kinematic gps … university of calgary quality control for...

UCGE Reports Number 20042

Quality Control for Differential Kinematic GPS Positioning

(URL: http://www.geomatics.ucalgary.ca/links/GradTheses.html)

Department of Geomatics Engineering

Ga

Aug

Calgary, A

By

ng Lu

ust 1991

lberta, Canada

THE UNIVERSITY OF CALGARY

QUALITY CONTROL FOR DIFFERENTIAL

KINEMATIC GPS POSITIONING

BY

GANG LU

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE IN ENGINEERING

DEPARTMENT OF SURVEYING ENGINEERING

CALGARY, ALBERTA

AUGUST, 1991

GANG LU 1991

ii

ABSTRACT

Kinematic differential GPS positioning is used to achieve accurate, real-time positions

and velocities of a moving platform on land, in the air and at sea. In a dynamic environment,

however, the deviations or biases from the assumed system and observation models are often

significant. This results in a degradation of the accuracy and reliability of the estimated positions

and velocities. Therefore, quality control methods are inevitably needed in a real-time

positioning system to insure the system is functioning properly. In this thesis, the statistical

quality control methods for use in kinematic GPS positioning are investigated. The general

recursive formulas for bias influence analysis and reliability analysis in Kalman filtering are

derived and the bias influence characteristics and the minimum detectable bias (MDB) values of

some typical biases in differential kinematic GPS positioning are studied. A real-time statistical

testing and implementation procedure for use in kinematic GPS positioning is given based on

the state space two-stage Kalman filtering technique, hypothesis testing theory and reliability

analysis. This procedure is for the detection of common biases such as cycle slips in carrier

phases and outliers in phase rates and pseudoranges, and for the elimination of their influences

on the kinematic GPS positioning results. All the derived formulas and algorithms were

implemented in a software package called QUALIKIN on a 386 micro-computer. The

application of the statistical quality control methods and testing of the software was performed

on two GPS data sets collected in land semi-kinematic mode over a well-controlled traverse.

Analysis of the results indicates that, with a data rate of one second or higher, the testing

procedure developed herein can correctly detect, identify and estimate carrier phase cycle slips

between consecutive epochs occurring at the one cycle level on multiple satellites under low and

medium vehicle dynamics. The detection and adaptation of phase rate and pseudorange outliers

is also possible. The magnitude of the detectable outliers is dependent on the corresponding

measurement accuracy, satellite geometry and the number of the outliers present. Further studies

iii

of quality control methods are recommended. Among them are the extension of the testing

procedure to incorporate the system bias detection and adaptation for high dynamic surveying

and navigation systems, application of the theory to integrated systems such as GPS/INS, and

investigation of the carrier phase ambiguity initialization methods in the kinematic mode.

iv

ACKNOWLEDGMENTS

I would like to express my deep gratitude to my supervisor, Professor Gerard Lachapelle,

for his continuous support and encouragement throughout the course of my graduate studies. His

advice and guidance were essential for the completion of this thesis.

Special thanks go to Dr. M.E. Cannon and Dr. Ming Wei, for their willingness to discuss

GPS data processing techniques and problems related to my research. Mr. Bryan Townsend is

sincerely thanked for sharing his knowledge in GPS and computer systems. He and Binbin She

are also thanked for helping me in the field tests. Ms Caroline Erickson is gratefully

acknowledged for carefully proofreading the manuscript. Finally, I wish to extend my

appreciation to the Faculty, staff and fellow graduate students of the Department of Surveying

Engineering for making my studies productive and enjoyable.

Financial support for this research came from The University of Calgary in the form of

Graduate Teaching and Research Assistantships and the Helmut Moritz scholarship fund. Their

contribution is gratefully acknowledged.

v

TABLE OF CONTENTS

Page

ABSTRACT .............................................................................................................. iii

ACKNOWLEDGEMENTS ............................................................... ............................ v

TABLE OF CONTENTS .................................................................. ............................. vi

LIST OF TABLES .......................................................................... ............................... ix

LIST OF FIGURES ........................................................................ ................................ x

NOTATION ................................................................................. .................................. xii

CHAPTER

1 INTRODUCTION ...................................................................... .............................. 1

1.1 Background ....................................................................... .............................. 1

1.2 Outline of Thesis .................................................................... ........................ 3

2 THEORY OF QUALITY CONTROL IN KALMAN FILTERING .............. .......... 4

2.1 General Description ......................................................................................... 5

2.2 Two-stage Kalman Filter ................................................................................. 7

2.2.1 Bias-free Kalman Filtering ............................................. ..................... 8

2.2.2 Two Step Estimation of State Vector and Bias Vector .............. ......... 10

2.3 Testing Procedures and Reliability Measures in Kalman Filtering ......... ........ 15

2.3.1 Testing Statistics for Biases ........................................... ..................... 15

2.3.2 Reliability Measures in Kalman Filtering ............................ ............... 19

2.3.3 Testing Procedures ..................................................... ......................... 21

3 KINEMATIC DIFFERENTIAL GPS POSITION AND VELOCITY ESTIMATION MODEL ............................................................... ........................... 25

3.1 System Models ................................................................... ............................. 26

vi

3.1.1 Constant Velocity Model ..................................................................... 26

3.1.2 Constant Acceleration Model .............................................................. 28

3.2 Observation Model ............................................................... ........................... 30

4 RELIABILITY ANALYSIS IN KINEMATIC DIFFERENTIAL GPS SURVEYS ....................................................................... ................................ 34

4.1 Bias Influences on Position and Velocity Estimation in Kinematic Differential GPS Surveys ......................................... ..................... 34

4.2 MDB in Kinematic Differential GPS Surveying Model ..................... ............ 45

5 CARRIER PHASE AMBIGUITY INITIALIZATION FOR DIFFERENTIAL KINEMATIC GPS SURVEYING ............................. ................. 51

5.1 Phase Ambiguity Initialization in State Mode ................................. ............... 52

5.2 Phase Ambiguity Initialization in Kinematic Mode .......................... .............. 54

5.2.1 Ambiguity Function .................................................... ........................ 55

5.2.2 Results of Phase Initialization by Ambiguity Function ....................... 58

6 TESTING AND RESULTS ........................................................... .......................... 64

6.1 Description of the Program QUALIKIN ...................................... ................... 64

6.2 Description of the Field Tests .................................................. ....................... 68

6.3 Positioning Results .......................................................................................... 70

6.3.1 Day 222 Results ................................................................................... 70

6.3.2 Day 121 Results ................................................................................... 74

6.4 Detection and Adaptation for Simulated Biases .............................................. 75

6.4.1 Detection and Adaptation for Simulated Cycle Slips .......................... 76

6.4.2 Detection and Adaptation for Simulated Outliers in Phase Rate Observations ........................................................................................ 82

6.4.3 Detection and Adaptation for Simulated Outliers in Pseudoranges Observation............................................. ............................................. 84

7 CONCLUSIONS AND RECOMMENDATIONS ................................................... 86

REFERENCES ............................................................................................................... 90

vii

LIST OF TABLES

Table Page

5.1 Resolved Initial Phase Ambiguities by the AFM Using 5 Epochs of Observations from the Data Set of Day 121, 1991 ......................... ................. 59

5.2 Resolved Phase Ambiguities by the AFM in Kinematic Mode Using One Epoch of Observations on Day 222, 1990 ........................................... 60

5.3 Resolved Phase Ambiguities by the AFM in Kinematic Mode Using Two Epochs of Observations on Day 121, 1991 ....................... ................. 62

6.1 Observation Information for the Semi-Kinematic Tests ....................... ................ 69

6.2 Results of Cycle Slip Detection and Estimation for Simulated Cycle Slips in the Data Set of Day 121, 1991 (1 Second Data Rate) ....................... ................ 77

6.3 Results of Cycle Slip Detection and Estimation for Simulated Cycle Slips in the Data Set of Day 222, 1990 (4 Seconds Data Rate) ...................................... 79

6.4 Results of Bias Detection and Estimation for Simulated Phase Rate Outliers in the Data Set of Day 121, 1991 (1 Seconds Data Rate) ........................ 83

6.5 Results of Bias Detection and Estimation for Simulated Pseudorange Outliers in the Data Set of Day 121, 1991 (1 Seconds Data Rate) ........................ 85

viii

LIST OF FIGURES

Figure Page

2.1 Process of Quality Control in Dynamic Systems ............................... ................... 5

2.2 Flowchart of the Testing Procedure for Use in Kalman Filters ............................. 24

4.1 Dilution of Precision for Part of the Trajectory of Day 222, 1990 ........................ 36

4.2 Cycle Slip Influences on Positions and Velocities. Six Satellites (SVs 2, 6, 9, 11, 13, 18) are available simultaneously. SV 11 is assumed to have one cycle slip (19.02) starting from 494132 seconds. The data interval is 4 seconds ................................................................ .............................. 37

4.3 Cycle Slip Influences on Positions and Velocities. Six Satellites (SVs 2, 6, 9, 11, 13, 18) are available simultaneously. Two satellites (SVs 11, 18) are assumed to have two cycles slip starting from 494132 seconds. The data interval is 4 seconds ......................................... ....................... 38

4.4 Pseudorange Outlier Influences on Positions and Velocities. Six Satellites (SVs 2, 6, 9, 11, 13, 18) are in view simultaneously. A single outlier of 20 m is assumed in the pseudorange of SV 11 at 494132 seconds. The data interval is 4 seconds ..................................................... .......................... 40

4.5 Phase Rate Outlier Influences on Positions and Velocities. Six Satellites (SVs 2, 6, 9, 11, 13, 18) are available simultaneously. A single outlier of 0.5 m s-1 in the phase rate occurs on SV 11 at 494132 seconds. The data interval is 4 seconds ..................................................... .................... 42

4.6 System Bias Influences on Predicted Positions and Velocities. An acceleration bias of 2 m s-1 occurs on the system state Vn at 6 consecutive epochs from time 494132 to 494152. The data interval is 4 seconds ....... ........... 44

4.7 Local MDB Improvement in Double Difference on Carrier Phase Observations on SV 02 with the Different Satellite Coverages ............... ............. 47

4.8 GDOP Improvement with the Different Satellite Coverages .................. .............. 47

4.9 Local MDB Improvements in Double Difference on Phase Rate Observations on SV 02 with the Different Satellite Coverages ............... ............. 48

4.10 Local MDB Improvements in Double Difference Pseudorange Observations on SV 02 with the Different Satellite Coverages ............... ............. 49

6.1 Flowchart of the Main Processor of the Program QUALIKIN ............... .............. 66

ix

6.2 Sketch of the Test Traverse Control Points ........................................................... 68

6.3 Position Differences with the Control Points for Day 222, 1990 .......................... 71

6.4 Differences Between QUALIKIN and Ashtech KINSRVY for Kinematic Results of Day 222, 1990 ..................................................................... 73

6.5 Position Differences with the Control Points for Day 121, 1991 .......................... 74

6.6 Comparison of the Results with and without Cycle Slip Adaptation for the Single Cycle Slip Case ............................................................................... 81

x

NOTATION

i) Symbols

an north acceleration

ae east acceleration

ah up acceleration

b bias vector

bo minimum detectable bias vector with power 1-o

B, C influence matrices that determine how the bias vector b enters into the

Kalman system and observation equations, respectively

c speed of light

d degree of freedom in chi-square distribution, number of biases

e observation noise

ei phasor or complex vector ei = cos + i sin

H design matrix

Ho null hypothesis

Ha alternative hypothesis

Hk Kalman gain matrix

m Dimension of the innovation vector at each epoch, or

number of update observations at each epoch

N carrier phase ambiguity

N(0, Q) normal probability distribution with expectation 0 and covariance

matrix Q

p pseudorange observation (m)

P Kalman state vector covariance matrix

xi

Q Kalman filter process noise covariance matrix

Qv covariance matrix of innovation vector

R Kalman filter observation noise covariance matrix

R earth radius

S sensitivity matrix related to the innovation vector

TG global model test statistic fGT global failure test statistic

f_LGT local failure test statistic Lgt local identification test statistic

U sensitivity matrix related to the predicted Kalman state vector

V sensitivity matrix related to the updated Kalman state vector

v innovation vector, or predicted residual vector

Vn north velocity

Ve east velocity

Vh up velocity

w process noise

x state vector

z observation vector

significance level or Type I error

inverse of correlation time of the stochastic Markov process

Type II error

1- power of test

non-centrality parameter in chi-square distribution

L1 GPS carrier wavelength

o critical value of which satisfies the given power 1-o and

significance level o

xii

Kalman filter transition matrix

carrier phase observations (m)

2(d,) chi-square probability distribution

geodetic latitude

geodetic longitude

h geodetic height

range from receiver to satellite

N double difference carrier phase ambiguity

ii) Defined Operators

(+) Kalman update

(-) Kalman prediction

derivative with respect to time

HT matrix transpose

Q-1 matrix inverse

single difference between receivers

single difference between satellites

deviation in

correction to

x) bias-free or bias-ignored Kalman filter estimated value

x~ bias-corrected Kalman filter estimated value

| . | norm operator

~ distributed as

xiii

iii) Acronyms

AFM Ambiguity Function Method

AVL Automatic Vehicle Location

C/A code Clear / Acquisition code

DD Double Difference

DGPS Differential GPS

GDOP Geometry Dilution of Precision

GPS Global Positioning System

GMT Global Model Test

HDOP Horizontal Dilution of Precision

INS Inertial Navigation System

LBNR Local Bias-to-Noise Ratio

LMT Local Model Test

MDB Minimum Detectable Bias

P code Precise code

RMS Root Mean Square

SD Single Difference

SV Space Vehicle

TD Triple Difference

VDOP Vertical Dilution of Precision

1

CHAPTER 1

INTRODUCTION

1.1 BACKGROUND

Kinematic surveying with the Global Positioning System (GPS) plays a key role in a

number of positioning activities, such as precise navigation, airborne gravimetry and

gradiometry, airborne photogrammetry and Automatic Vehicle Location (AVL). With the full

deployment of 24 GPS satellites in the near future and the improved receiver technologies

(e.g., all-in-view satellite receivers), kinematic GPS surveys, which include the determination

of position, velocity or even acceleration of a moving platform, may become a routine job for

geodesists and surveyors. As we know from conventional geodetic surveys, the quality

assessment of surveying results generally involves accuracy analysis, reliability analysis and

statistical testing of the estimated quantities and adequacy of the adjustment model. Accuracy

analysis deals with the propagation of random errors through the geometric strength of the

network or the adjustment model, while reliability analysis and statistical testing,which are the

primary elements of quality control, deal with the self-check ability of the model or system to

blunders or biases that occur in observations or in the systems. Up to now, extensive

investigations and tests have been done in the accuracy analysis of kinematic differential GPS

surveys. The results of 0.2m (standard deviation) in kinematic positioning and 0.05m/s

2

(standard deviation) in velocity estimation have been made by using both pseudorange and

carrier phase observations (Lachapelle et al, 1989; Schwarz et al, 1989; Cannon, 1987). The

reliability analysis and statistical testing in kinematic GPS surveys, however, is just at its initial

stage. Teunissen and Salzmann (Teunissen, 1990; Salzmann and Teunissen, 1989) introduced

the quality control concept of conventional geodetic surveys into kinematic surveying systems

and developed the general quality control theory based on the standard state space Kalman

filtering model and the statistical hypothesis testing. The first application and adaptation of

this theory to kinematic GPS positioning was given by Wei et al (Wei et al,1990) and Lu and

Lachapelle (Lu and Lachapelle,1990).

It is known that by using GPS pseudorange, carrier phase and phase rate (Doppler)

observations in a Kalman filter, the instantaneous position, velocity and even acceleration of a

moving platform can be determined (Schwarz et al,1989; Cannon,1990; Hwang and Brown,

1990). If the assumed dynamic and observation models are correct, the Kalman filter provides

the optimal position and velocity estimates in a statistical sense. In a dynamic environment,

however, the deviations or biases from the assumed models are often significant. For instance,

the loss/re-lock of a carrier phase signal on a satellite will cause the corresponding integrated

carrier phase observations to jump abruptly, which falsifies the measurement update model.

Likewise, a pre-determined constant velocity model for vehicle motion may be invalidated due

to high vehicle accelerations in some parts of the trajectory. Such deviations or biases will

certainly lead to some errors in the filtering results. Therefore, reliability analysis and real-

time testing and adaptation for possible biases is of great importance in kinematic GPS surveys

in order to prevent the degradation of position and velocity estimates.

3

1.2 OUTLINE OF THESIS

The primary objective of this thesis is to apply the quality control methods to kinematic

GPS surveys. The general formulas and procedures for reliability analysis and bias detection

and adaptation in Kalman filtering are derived. A software package, QUALIKIN, that

performs reliability analysis, real-time statistical testing and adaptation for possible biases in

pseudo-range, carrier phase and phase rate observations along with kinematic differential GPS

positioning and/or semi-kinematic positioning, is developed.

Chapter 2 describes the quality control theory of the state space Kalman filtering.

Firstly, a more general recursive bias influence formulation on filtered quantities is introduced

by using a two-stage Kalman filter method. Then, a real-time statistical testing procedure for

use in the Kalman filter is given based on the hypothesis testing theory and the Minimum

Detectable Bias (MDB) concept. The system model and the observation model for kinematic

positioning used in this research are outlined in Chapter 3. Chapter 4 investigates the influence

characteristics of different biases (e. g., cycle slips) on kinematic positioning results and the

corresponding minimum detectable biases. In Chapter 5, methods for carrier phase ambiguity

initialization are reviewed. In particular, the ambiguity function method (Mader, 1990) is

discussed and tested for phase ambiguity resolution in kinematic mode. The processing results

of real data sets and the applicability of statistical methods for detecting and correcting biases

in observations are listed in Chapter 6. Some conclusions and recommendations are given in

Chapter 7.

4

CHAPTER 2

THEORY OF QUALITY CONTROL IN KALMAN FILTERING

The quality control method, i.e. reliability analysis and statistical testing of the

adequacy of adjustment models, has been widely used for network design and data processing

in conventional geodesy, photogrammetry and surveying (Baarda, 1968; Li, 1986; Pelzer,

1986). In systems engineering, it is often called fault detection, diagnosis and adaptation

(Basseville,1988; Frank,1990). With the advent of the global positioning system (GPS),

kinematic surveying is now becoming a recognized surveying method on land, in the air and on

the sea. Therefore, it is natural and important to introduce and apply the quality control

method to this new kinematic surveying method to insure the correctness of the results

obtained.

In this chapter, the general concept of quality control in dynamic systems engineering is

reviewed first. Then, two-stage Kalman filter formulas are derived, which are useful for

analyzing the bias influence on Kalman filter output quantities. And finally, the formulation of

testing procedures and reliability analysis is presented.

5

2.1 GENERAL DESCRIPTION

The process of quality control or fault detection, diagnosis and adaptation in dynamic

systems essentially consists of three stages (Basseville,1988; Teunissen,1990): residual

generation, decision making and adaptation or estimation, as shown in Fig. 2.1.

system and sensor outputs

residual generation

calculation of decision statistics

failure decision and identification

adaptation measures

final output

residual generation decision making adaptation

Fig. 2.1 Process of Quality Control in Dynamic Systems

The residual generation is based on the knowledge of the normal behaviour of the

system and the characteristics of the failures. For detection purposes, constructed residuals

should significantly and quickly reflect the influences of possible failures or biases which

occurred in the analyzed system, and remain unbiased, normally close to zero, in the absence

of failures or biases. In kinematic surveying systems like GPS and Inertial Navigation Systems

(INS), Kalman filtering is widely used for data processing. In this case, one suitable choice of

residual generation is the filter output of innovation sequence, i.e. predicted residuals (Willsky,

1976; Teunissen, 1990). The generation of innovation sequence and its properties are given in

Section 2.2.

6

Decision making is the construction of decision functions or test statistics based on the

generated residuals. Decision functions or statistics are then calculated using the residuals to

determine if any failure has occurred in the system. In some cases, failure identification or

isolation should also be carried out. For innovation-based detection systems like Kalman

filters, a number of statistical tests to be performed on the innovations have been suggested.

Among them are the Chi-square statistic and the Generalized Likelihood Ratio (GLR) test

(Teunissen, 1990; Willsky, 1976; Mehra and Peschon, 1971). Due to its simplicity and easy

implementation, the real-time Chi-square test is used in this research for detecting and isolating

the possible biases (e.g., cycle slips on some satellites) in the kinematic GPS surveying model.

Adaptation is system reconfiguration to accommodate the failure, or to estimate the

failure or bias and then correct its influences on system outputs. Reconfiguration measures are

often used in a system with a high degree of parallel hardware redundancy (e.g., voting

system), while analytical correction methods are usually adopted in a single feedback system.

For kinematic surveying with a single GPS system, only real-time analytical correction

methods are possible. For this purpose, two-stage Kalman filter techniques are derived for

real-time bias estimation and correction.

Another important aspect of quality control is the reliability analysis of the systems.

This concept was mainly developed by geodetic scientists (Baarda, 1968; Teunissen, 1990) and

has not yet been discussed in system engineering literature. Generally speaking, reliability is

concerned with the self-checking ability of the system model for possible biases and the effects

of undetectable biases on the estimated results. One of the numerical measures for reliability

in dynamic systems is the minimum detectable bias (MDB) (Teunissen, 1990), which is the

minimum bias value that can be detected with a certain probability by a specified bias testing

statistic in an level test. The importance of reliability analysis lies in the design stage of a

dynamic system. By examining the reliabilities of different design schemes, an efficient

7

system with sufficient control on the presence of certain biases can be obtained. In kinematic

GPS surveys, for instance, the reliability analysis can be conveniently used for investigating

the minimal detectable cycle slips occurring on a certain satellite and the cycle slips effect on

position and velocity determination. This problem and the application of reliability analysis in

kinematic differential GPS surveys will be discussed in Chapter 4.

2.2 TWO-STAGE KALMAN FILTER

Under the nominal conditions that the system and observation model and the statistical

model are correct, the Kalman filter provides estimates of the state vector which are unbiased

and of minimum variance. Due to the dynamic environment and possible failures of some

system components, deviations or biases from the predetermined filter model are often

encountered in practice. An important class of biases, which is suitable to represent cycle slips

in carrier phase measurements and outliers in pseudorange and phase rate observations, are the

constant biases with unknown magnitudes. Usually, there are two ways to treat these kinds of

biases when they are detected in the system dynamic model or the observation model. Firstly,

one can augment the state vector of the original model by adding components to represent the

bias terms. The filter then estimates these terms as well as the original states. This

mechanism, for example, is implemented in the well-known program SEMIKIN for estimating

the cycle slips in carrier phase measurements (Cannon, 1990). Secondly, the bias terms can be

estimated separately from the original filter states by using the bias-free, i.e. the nominal

Kalman filtering results. This is the so-called two-stage Kalman filter (Friedland, 1969;

Ignagni, 1981). Mathematically, these two methods are equivalent. However, the second

method has the advantages of being computationally efficient and convenient for both bias

influence analysis and reliability analysis of filtering results. Therefore, it is used in this

research for reliability analysis and processing software development.

8

2.2.1 Bias-free Kalman Filtering

The general Kalman filter equations are well documented in Gelb (1974). For a system

described by the following equations

xk = k xk-1 + wk , wk ~ N(0, Qk ) (2.1)

zk = Hk xk + ek , ek ~ N(0, Rk ), (2.2)

the optimal estimate of the state vector xk and its covariance matrix are given by

Prediction: xk(-) = k xk-1(+) (2.3)

Pk(-) = k Pk-1(+) kT + Qk (2.4)

Update: xk(+) = xk(-) + Kk [ zk - Hk xk(-) ] (2.5)

Pk(+) = [ I - Kk Hk] Pk(-) (2.6)

Kk = Pk(-)HkT[ HkPk(-)Hk

T + Rk ]-1 (2.7)

where (-) denotes the predicted quantities, i.e., before update,

(+) denotes the updated quantities,

xk is the state vector,

k is the transition matrix,

Pk is the state covariance matrix,

Kk is the Kalman gain matrix,

Qk is the covariance matrix of the system process noise wk ,

Rk is the covariance matrix of the observation noise ek ,

Hk is the design matrix,

9

and zk is the observation vector for updating.

If the equations (2.1) and (2.2) correctly specify the underlying system model and

observation model, the Kalman algorithm from equations (2.3) to (2.7) gives the minimum

variance, unbiased estimate of the state vector x at each epoch k. In this case, the innovations,

i.e. the predicted residuals vk described as

vk = zk - Hk xk(-) (2.8)

have a zero mean, Gaussian white noise sequence with covariance matrix

Qvk = Rk + Hk Pk(-)HkT (2.9)

(Mehra and Peschon, 1971; Willsky, 1976; Teunissen, 1989). Any deviation or bias arising

from the system model (2.1) or from the observation model (2.2) may cause the innovation

sequence (2.8) to depart from its zero mean and whiteness properties. This makes the

innovation sequence an ideal generated residual sequence in Kalman filtering for detecting the

abnormal behaviours of the system.

2.2.2 Two Step Estimation of State Vector and Bias Vector

In the presence of constant biases with unknown magnitudes in the functional model of

Kalman filtering, equations (2.1) and (2.2) can be reformulated as:

xk = k xk-1 + Bk b + wk , wk ~ N(0, Qk ) (2.10)

zk = Hk xk + Ck b + ek , ek ~ N(0, Rk ), (2.11)

where b is the constant bias vector. The matrices Bk and Ck determine how the components

of the bias vector b enter into the dynamics and observations respectively. In the case when

10

only observations have biases, Bk = 0. Similarly, if only the dynamics have biases, Ck = 0.

If no biases present in the model, Bk = Ck = 0.

In one step (simultaneous) estimation, the bias vector b in (2.10) and (2.11) is

appended to the original state vector x with the dynamics bk = bk-1 to form a new state

vector y = ( x, b ). Then, the new state vector y is estimated by the conventional Kalman

algorithm (2.3) to (2.7) based on the augmented model

yk = Fk yk-1 + Gwk , wk ~ N(0, Qk ) (2.12)

zk = Lk yk + ek , ek ~ N(0, Rk ) (2.13)

with the partitioned matrices

Fk =

k Bk

0 I , G =

I

0 , Lk = ( Hk , Ck ) . (2.14)

This method preserves the mathematical simplicity, but has the disadvantages that it increases

the dimension of the state vector and is not suitable for reliability analysis and bias influence

analysis in Kalman filtering.

The two step estimation of states and biases overcomes the disadvantages in the above

one step estimation procedure. The basic idea for two step estimation of states x and biases b

is as follows. In the first step, we ignore the bias terms in (2.10) and (2.11), and perform the

conventional bias-free Kalman filtering procedure. Due to the omission of bias terms, we then

obtain the biased state estimates, xk(+) and xk(-) , and the biased innovations vk that contain

the information on the biases. In the second step, the bias vector b is estimated by using the

innovations obtained in the first step, and the biased state estimates, xk(+) and xk(-) , are

corrected for the bias influences. Mathematically, two step estimation of states and biases is

equivalent to one step estimation. The proof is given by Friedland (1969) and Ignagni (1981).

11

Here, only an intuitive derivation is given and the emphasis is placed on the computational

aspects.

a. Bias influences on bias-free state estimates

If the bias vector were perfectly known, as is the case when we analyse the influence of

a specified bias vector on the Kalman filtering results, the optimal estimates of the state vector

x in equations (2.10) and (2.11) would be

xk(-) = kxk-1(+) + Bkb (2.15)

xk(+) = xk(-) + Kk( zk - Hkxk(-) - Ckb ) , (2.16)

while the innovations with known biases would be

vk = zk - Hkxk(-) - Ckb , (2.17)

where b is the true value of the constant bias vector, Kk is the Kalman gain matrix defined by

(2.7) of the bias-free model, and xk(-) and xk(+) are the bias-corrected Kalman state estimates

before and after update, respectively.

Since the Kalman filter estimator is a linear estimator and the bias influence on the

filter estimates, as reflected in (2.16) and (2.17), is also linear in nature, we can formally write

the relationships between the bias-ignored estimates xk(-) and xk(+) and bias-corrected

estimates xk(-) and xk(+) as (Friedland, 1969):

xk(-) = xk(-) + Ukb (2.18)

xk(+) = xk(+) + Vkb . (2.19)

For the same reason, the relationship between the corresponding bias-ignored innovations vk

and the bias-corrected innovations vk can be written as

12

vk = vk + Skb , (2.20)

where Uk , Vk and Sk are called sensitivity matrices . Equations (2.18), (2.19) and (2.20)

indicate that in the presence of biases, the optimal Kalman filter state (bias-corrected)

estimates can be obtained by adding corrections to the bias-free or bias-ignored Kalman filter

estimates.

Substituting equations (2.15), (2.3) and (2.19) into (2.18), we obtain the recursive form

for computing Uk :

Uk = k Vk-1 + Bk (2.21)

Similarly, the recursive forms for Vk and Sk can be derived as:

Vk = Uk - Kk Sk (2.22)

Sk = Hk Uk + Ck .

(2.23)

The initial value for the recursive computations of Uk , Vk and Sk starting at epoch k is

Vk-1 = 0 in equation (2.21). It is noted that the computations of Uk , Vk and Sk are

independent of the measurements. They can be computed beforehand using the specified

dynamics model, observation model and bias type. Equations (2.21), (2.22) and (2.23) are of

great importance in bias influence analysis. If a bias occurs in the system, its influence on the

bias-free Kalman filtering estimates xk(-) , xk(+) and vk , as shown in (2.21), (2.22) and

(2.23), are -Uk b, -Vk b and Sk b respectively. In kinematic GPS surveys, the most likely

biases are carrier phase cycle slips, outliers in pseudorange and phase rate observations, and

the deviations of the vehicle motion from the assumed dynamics. Their influence on kinematic

GPS position and velocity determination is to be investigated in Chapter 4.

b. Bias estimation

13

In real data processing, the true value of the bias vector b is usually unknown. It is

required that the bias vector be estimated in real time in order to correct its influences on the

filtering results. This can be achieved by utilizing the equation (2.20). Rewriting (2.20) in the

form:

vk = Skb + vk , (2.24)

the bias-free Kalman filter innovations vk can then be considered as a quasi-measurement

vector, Sk as a design matrix and vk, the bias-corrected innovations, as the quasi-measurement

noise. Based on the properties of Kalman filtering and equation (2.17), the bias-corrected

innovation sequence vk is a white noise sequence with the covariance matrix at each epoch:

Qvk = Hk Pk(-) HkT + Rk, (2.25)

where Pk(-) is given by equation (2.4).

From the above discussions, it is obvious that the bias vector b can be recursively

estimated with a Kalman filter algorithm or an equivalent sequential least squares method

using the measurement equation defined by (2.24) with the measurement white noise

covariance matrix (2.25). The Kalman filter algorithm for recursive estimation of the bias

vector b has the form

b (-)(k) = b (+)(k-1) ( b0 = 0 ) (2.26)

Pb(-)(k) = Pb

(+)(k-1) ( Pb(0) given ) (2.27)

b(+)(k) = b (-)(k) + Kb(k)[ vk - Skb

(-)(k) ] (2.28)

Pb(+)(k) = [ I - Kb(k)Sk ] Pb

(-) (k) (2.29)

Kb(k) = Pb(-) (k) Sk

T [ Sk Pb(-) (k) Sk

T + HkPk(-)HkT + Rk ]-1 (2.30)

14

This constitutes the second stage of two-stage Kalman filtering, i.e. bias estimation. It is noted

from equation (2.30) that we need to invert a matrix of dimension that is equal to the number

of observations at each epoch. In the kinematic GPS positioning model, the number of biases

is usually smaller than the number of observations at each epoch. Therefore, the equivalent

Bayes form or phase expressions (Krakiwsky, 1990) are more favourable than the Kalman

algorithm given by (2.26) through (2.30), since the Bayes form inverts a matrix of dimension

that is equal to the number of biases.

c. Bias influence correction and adaptation

According to equations (2.18) and (2.19), once the bias estimates are obtained, the bias-

ignored Kalman filter estimates, xk(-) and xk(+) , and their covariance matrices can be

recursively corrected for the bias influences by:

xkc(-) = xk(-) + Uk b

(-)(k) (2.31)

xkc(+) = xk(+) + Vk b

(+)(k) (2.32)

Pkc(-) = Pk(-) + UkPb

(-)(k)UkT (2.33)

Pkc(+) = Pk(+) + VkPb

(+)(k)VkT (2.34)

Pxb(-) (k) = UkPb

(-)(k) (2.35)

Pxb(+)(k) = VkPb

(+)(k) (2.36)

where xkc(-) , xkc(+), Pk

c(-), Pkc(+) are the optimal unbiased state estimates and their

corresponding covariance matrices, which have taken into account the influence of the biases

in the models. In the derivation of (2.33) to (2.36), the fact that the state vector x initially and

at all subsequent epochs is uncorrelated with the biases b has been used.

15

2.3 TESTING PROCEDURES AND RELIABILITY MEASURES IN KALMAN

FILTERING

2.3.1 Testing statistics for biases

Failure detection and identification are the most important and yet the most difficult

aspects in the quality control of Kalman filtering. Generally speaking, failures or

malfunctioning of a Kalman filter may be caused by two error sources. One is the errors

arising from the function models, such as the unmodelled biases in the measurements and

system dynamics. The other source is the errors arising from the stochastic models defined

mainly by the covariance matrices of the system process noise wk and the measurement noise

ek .

In differential (double difference) kinematic GPS surveying, the functional model

errors, such as the cycle slips in carrier phases and the outliers in pseudoranges and phase rates,

are frequent and have severe influences on position and velocity determination. The stochastic

model errors, relatively speaking, are much less significant than the functional model errors.

The measurement noise in a surveying session is mainly specified by the instrumentation and

environment and can be considered as a constant (Lachapelle, 1990). The system process

noise, on the other hand, is determined by the vehicle motion. In a low or medium dynamic

surveying system, the first-order Gaussian-Markov model of constant velocity or constant

acceleration of the vehicle motion seems justified (Schwarz et al, 1989; Cannon, 1990). In

practical computations, we usually use slightly larger values for measurement variance and

process noise spectral density to cope with the unpredictable statistical behaviors of the

system. Therefore, in the following the statistical models of the Kalman filter are considered

correct and the focus is placed on the detection and isolation of biases in the function models.

16

In order to develop the statistics for bias detection, we first define a vector consisting of

n+1 epochs of innovations of equation (2.8)

v = ( vl , vl+1 , vl+2 , ......, vl+n )T . (2.37)

As we know from section 2.2, under normal conditions when the Kalman filter models are

specified correctly and no biases are present, the innovation sequence is a zero mean,

Gaussian white noise sequence. In this case, vector v has a normal distribution

v ~ N(0, Qv ), (2.38)

where Qv is a block-diagonal matrix since the innovations are uncorrelated from epoch to

epoch (Teunissen and Salzmann, 1989). However, if a bias vector b of dimension d is present

in the functional models, the zero mean of the innovations at each epoch is no longer holds.

Instead, the innovation sequence is biased, as known from (2.20), by a value Sk b. This leads

to

v ~ N(v, Qv ) (2.39)

with v = Sv b , (2.40)

Sv = ( SlT , Sl+1

T , ......., Sl+nT )T , (2.41)

where Sv is a (i=l

l+nmi )-by-d matrix and mi is the dimension of vi .

Thus, the problem of detection of the bias vector b can be formulated as the testing of the null hypothesis Ho against the alternative hypothesis Ha :

H0 : v ~ N(0, Qv) versus Ha : v ~ N(v, Qv) . (2.42)

17

The testing statistic for the above problem (2.42) is well documented in a number of

publications (Teunissen, 1986, 1990; Rao,1973 ) and is given by

TG = vT Qv-1 Sv (SvT Qv-1 Sv)-1 SvT Qv-1 v . (2.43)

Since the matrix Qv is block diagonal, (2.43) can be further reduced to the summation form

based on (2.37), (2.41):

+

=

+

=

+

=

=n1

1ii

1v

Ti

n1

1i

1i

1v

Ti

n1

1i

Ti

1v

TiG )vQS()SQS()vQS(T iii

. (2.44)

Under the null hypothesis Ho , TG has a central 2 -distribution, i.e.

TG ~ 2( d, 0 ) under Ho , (2.45)

while under the alternative hypothesis Ha , TG has a non-central 2 -distribution, i.e.

TG ~ 2( d, ) under Ha , (2.46)

where d is the degrees of freedom which is equal to the number of biases and is the non-

centrality parameter which is defined by

+

=

=n1

1ii

1v

Ti

T )SQS(i

bb (2.47)

The null hypothesis that there are no such biases b present in the model is accepted if TG is

less than the upper -percentage point of the central 2 -square probability distribution

2( d, 0 ) , i.e.

Ho is accepted if TG 2( d, 0 ) . (2.48)

Otherwise, Ho is rejected in favour of Ha , meaning that the specified bias vector b is

statistically significant under the significance level or risk level .

18

The test statistic of (2.44) can be sub-divided into two cases. They are the so-called

local model test (LMT) and global model test (GMT) (Teunissen and Salzmann, 1989). If

only the current one epoch innovation is used to form the statistic (2.44), it is called a local

test. This corresponds to n=0 in (2.44). Otherwise, it is a global test which means that more

than one epoch of innovations have been included in the computation of the statistic TG . The

main advantage of the global test is the better detection power for systematic model bias trend

(Teunissen and Salzmann, 1989; Lu and Lachapelle, 1990). But we pay for it with increased

computational complexity and time.

In real-time data processing, once a bias vector b is detected and estimated, its effects

on the bias-free Kalman filter results should be immediately corrected in order to prevent

further deterioration in subsequent epochs. This can be done by the algorithms of the two-

stage Kalman filter given in the previous subsection 2.2.2.

2.3.2 Reliability measures in Kalman filtering

When applying the statistical testing for certain postulated biases, we are likely to make

two types of errors. A Type I error is the rejection of the null hypothesis Ho when it is true.

The probability of making this type of error (false alarm) is the test significance level or risk

level . A Type II error is the acceptance of Ho when the alternative hypothesis Ha is

actually true. The probability of making type II errors is denoted by , which is related to the

non-centrality parameter of the alternative test distribution and the given significance level

. Unfortunately, type I and type II errors can not be minimized at the same time. They are

related through the non-centrality parameter of the alternative test distribution. By fixing any

two of , and , the third one can be computed. Generally, the larger the non-centrality

parameter of the alternative test distribution, the smaller the probability of type II error or the

19

larger the power 1- of the statistical test. In this case, we may ask a question on how large

the non-centrality parameter should be in order to satisfy a pre-determined power 1-o of the

statistical test associated with a significance level o . This readily brings us to the important

concept of reliability in geodetic science (Baarda, 1968; Pelzer, 1988, Teunissen, 1990).

Reliability is mainly concerned with the effects of possible biases in the model on the

estimated results and the ability of the redundant information in the model to check against the

model biases or misspecifications. In Kalman filtering, the influences of fixed biases on the

filtering results can be easily investigated by using two-stage Kalman filter equations (2.18),

(2.19) and (2.20), while the ability to detect the individual bias in the system, which is usually

termed internal reliability in surveying, is measured with the quantity of minimum detectable

bias (MDB).

Suppose o is the minimum value of the non-centrality parameter which satisfies the

given test power 1-o and significance level o . From (2.47), we immediately obtain the

minimum detectable bias (MDB) vector bo as:

+

=

=n1

1i0i

1v

Ti

T00 )SQS( i

bb . (2.49)

For a single bias, bo is reduced to a scale and (2.49) can be written as

+

=

=

n1

1ii

1v

Ti

00

)SQSi

b . (2.50)

Equation (2.50) shows that the magnitude of a single bias should at least reach bo in order to

be detected with the power 1-o in an o significance level test. This property is utilized in

the testing procedures to be described in the next subsection.

20

The minimum detectable bias vectors corresponding to the Local Model Test and the

Global Model Test are referred to as Local MDB and Global MDB vectors respectively. For

multiple biases, the quadratic form (2.49) describes a hyper-ellipsoid if o is held to a constant

value. The axes of this hyper-ellipsoid are the inverses of the square roots of the eigenvalues

of the matrix +

=

n1

1ii

1v

Ti )SQS( i

which has a dimension equal to the number of biases .

A number of tables have been provided for calculation of the non-centrality parameter

o (Caspary,1987; Baarda,1968) . For example, in a one-dimensional alternative hypothesis

test (2.46) with the degrees of freedom equal to one, o is equal to (4.13)2 when we set o

= 20% and o = 0.1% . The computation of Si and Qvi-1 , as known from the two-stage

Kalman filter algorithm, is independent of the actual measurements. Hence, the MDB of a

specified bias can be computed before the filter is actually implemented. This makes the MDB

a useful tool in Kalman filter design. An efficient filter design with required reliability control

on the postulated biases can be achieved by examining the different filter operation schemes.

2.3.3 Testing procedures

The test statistic (2.43) or (2.44) is aimed at detecting a specified bias vector b of

dimension d through the sensitivity matrix Si . In a particular application, however, we are

usually not sure beforehand when and what kind of biases will occur in the system. It is

therefore necessary to establish testing statistics to detect first whether there is a failure or

malfunctioning in the underlying filter operations or not. This can be realized by leaving the

bias vector b completely unspecified, that is, letting the dimension of the bias vector b equal to

the dimension of the innovations vector v (Teunissen and Salzmann, 1989). In this case, the

matrix Sv in (2.41) becomes a square invertable matrix. Thus, it can be eliminated from the

test statistic (2.43), which results in

21

+

=

+

=

==n1

1i0i

2n1

1ii

1v

Ti

1v

TfG Hunder)0,m(~vQvvQvT i

. (2.51)

Equation (2.51) is basically a failure or alarm test that tells us whether there is anything

wrong in the system or not. If n = 0 in (2.51), i.e. only the current one epoch innovations are

used, it is called a Local failure or Local alarm test, which has the form:

TG L-f = vviT Qvi-1 vi ~ 2(mi, 0) under H0 . (2.52)

Once the failure test (2.51) or (2.52) signifies a failure present in the system, the

diagnosis of the failure sources, i.e. bias identification or isolation process, is conducted. For

this purpose, an approach similar to data-snooping (Baarda,1968) is employed in this research.

That is, each individual postulated bias source is tested separately until all the possible bias

sources are examined. This corresponds to the case when the dimension of the bias vector b

in (2.40) is chosen equal to one at each time. Thus the test statistic (2.43) reduces to a one-

dimensional identification or slippage test (Teunissen and Salzmann, 1989):

0

2n1

1ii

1v

Ti

n1

1i

2i

1v

Ti

g Hunder)0,1(~sQs

)vQs(t

i

i=

+

=

+

=

, (2.53)

where si is a one dimensional vector and computed by equation (2.23). For instance, if one

suspects sensor failures or outliers in the Kalman filter measurement model, one can chooses

Ck in (2.11) or (2.23) as

Cjk = ( 0, ..., 0, 1, 0, ..., 0 )

T (2.54)

1 j mk

for j = 1, 2, .... mk . This means that each observation is tested in turn to search biases.

22

If n is set to zero in (2.53), i.e. only the current epoch innovations are used for bias

identification, it is called a Local identification and has the form:

tg L =

(siTQvi-1vi)2

siTQvi-1si ~ 2(1, 0) under H0

.

(2.55)

Research experiences (Lu and Lachapelle, 1990; Wei et al, 1990) have shown that the

local failure test (2.52) and the local identification test (2.55) are suitable for bias detection and

identification in differential kinematic GPS surveying, since the most common and severe

biases are cycle slips which are multiples of the carrier wavelength (19.02 cm). Therefore, the

local test statistics (2.52) and (2.55) are used in the developed testing procedures.

Unfortunately in some cases, especially in large or multiple bias situations, the

identification test (2.55) is too sensitive and may lead to many false bias alarms, signifying

more biases than there are actually present. To overcome or alleviate this problem, a new step

is introduced in the testing procedure by using the concept of minimum detectable bias (MDB).

That is, for a bias signified by (2.55), its estimation value is compared against its

corresponding MDB value. If the estimated value of the bias is larger than its corresponding

MDB value, then this bias is considered an actual bias present in the Kalman filter models,

otherwise it is considered as a false bias alarm and eliminated from the bias vector b. By using

this method, only the biases with more than the detection power 1-o are retained and

estimated.

Summarizing the above discussions, we arrive at the following testing procedures as

depicted in Figure 2.2, which can be executed in parallel with the real-time Kalman filter

algorithm.

23

Innovation sequence

Recursivecomputation quantities

from the last epoch

Failure detection by alarm test Failure ?

yes

Bias identification

Bias confirmation by MDB and re-estimation

Adaptation or correction for bias influences on filtering results and measurements

Bias estimation by two-stage Kalman filter

no

Biasesfrom the last

epochyes

no

Next epoch

Fig. 2.2 Flowchart of the Testing Procedure for Use in Kalman Filters

24

CHAPTER 3

KINEMATIC DIFFERENTIAL GPS POSITION AND

VELOCITY ESTIMATION MODEL

Generally speaking, there are two kinds of models used for combined processing of

pseudo-range, carrier phase and Doppler frequency (phase rate) observations in precise

kinematic differential GPS (DGPS) surveys. One is the complementary filter or batch-

solution (Seeber et al, 1986; Hwang and Brown , 1990), in which one type of observable (e.g.

pseudorange) is used for correcting the positions derived by another type of observable (e.g.

carrier phase). The advantage of this type of modelling is that it does not require assumptions

on the motion behaviour of the moving platform. However, this type of model cannot directly

output the velocity and acceleration estimates. The second type of model is the integrated

filter or state space Kalman filter model (Schwarz et al, 1989; Hwang and Brown, 1990), in

which all available observations are processed simultaneously through a Kalman filter that

describes the kinematic surveying system. This type of model has been widely tested under

different environments and has yielded very good position and velocity estimation results in

kinematic and semi-kinematic surveying (Schwarz et al, 1989; Cannon et al, 1990; Cannon,

1990; Cannon, 1991). Therefore, the integrated filter or state space Kalman filter model is

used in this research for kinematic GPS data processing.

25

3.1 SYSTEM MODELS

In kinematic GPS surveying, if the satellite positions are considered fixed, the status of

a moving vehicle can be described by the vehicle's position, velocity and acceleration in three-

dimensional space, through the use of kinematic motion equations in physics (Schwarz et al,

1989). Usually, the constant velocity or constant acceleration model is adopted in surveying

practice. The adequacy and accuracy of these models depends on the dynamics of the vehicle

motion and the measurement update interval t. A detailed investigation and comparison of

the model behaviours is given in Schwarz et al (1989). Experiences in this research show that

a one second or higher data output rate is preferred for bias detection purposes when the

constant velocity model or the constant acceleration model is used as the system model for

land vehicle kinematic surveying systems. With the improvement of GPS receiver technology,

most of the new receivers, such as,.the department's Ashtech LD-XII, have a one Hz or higher

data output rate. With such a short time interval between epochs, the constant velocity model

or the constant acceleration model seems accurate enough for most of the kinematic land

surveying tasks.

3.1.1 Constant Velocity Model

Using the satellite-receiver double difference observables of the GPS system as the

update measurements, the constant velocity model describing the vehicle motion consists of

three position states (, , h) and three velocity states (Vn , Ve , Vh ). Thus, the state

vector x6 of the constant velocity model reads

x6 = ( , , h, Vn , Ve , Vh )T , (3.1)

26

where is the latitude, the longitude, h the height, Vn the north velocity, Ve the east

velocity, and Vh the up velocity. Here we assume that before the kinematic run begins, the

carrier phase ambiguity parameters have been resolved correctly by some common method,

such as antenna swapping or occupying the initial baseline for about 10~15 minutes to allow

for a static solution. Therefore, the ambiguity parameters are held fixed and do not appear in

the state vector.

The states Vn , Ve , Vh in (3.1) can be assumed to behave as a random walk or a

first-order Gauss-Markov process, according to the vehicle motion. If the first-order Gauss-

Markov process is used to describe the behaviour of the three velocity states, the transition matrix x6

of the state vector x6 has the form:

x6(t) =

I CD

0 T , (3.2)

where all submatrices are diagonal and of dimension (3x3), which can be expressed as:

C(t)3x3 = diag( ci ) with ci =

1i

( 1 - exp(-it ) ) , (3.3)

T(t)3x3 = diag( ti ) with ti = exp( -it ) , (3.4)

D3x3 = diag( 1/R, 1/R cos, 1 ). (3.5)

where i is the inverse of the correlation time of the stochastic Markov process, t is the

update interval and R is the earth radius.

The covariance matrix Qk of the system process noise related to x6 can be computed

by simple numerical integration when the spectral density matrix of the system noise Q(t) is

specified (Gelb, 1974), i.e.

27

Qk = x6()Q()x6

T () d=0

t

.

(3.6)

In the constant velocity model, Q(t) usually has the form

Q(t) =

0 0

0 Q2(t) ,(3.7)

where Q2(t) = diag( qvn(t) , qve(t) , qvh(t) ) are spectral densities of the three velocity states.

3.1.2 Constant Acceleration Model

Similar to the constant velocity modelling of the vehicle motion, the constant

acceleration model uses the position states (, , h), velocity states (Vn , Ve , Vh ) and

acceleration states (an , ae , ah ) to describe the motion of the vehicle in 3-dimensional

space. This leads to the following filter state vector

x9 = ( , , h, Vn , Ve , Vh , an , ae , ah )T , (3.8)

where the three additional acceleration states an , ae , ah , corresponding to north, east and

up accelerations respectively, are added. In this model, the three acceleration states are

considered to be driven by Gaussian noise and to behave as a random walk or a Gauss-Markov

process depending on the vehicle dynamics. If the first-order Gaussian-Markov process is used

to describe the acceleration states, the transition matrix of state vector (3.8) reads

x9(t) =

I Dt FD

0 I C0 0 T

, (3.9)

28

where submatrices C, T and D are defined as in (3.3), (3.4) and (3.5). The submatrix F has the

form

F(t)3x3 = diag( fi ) with fi =

1i

2(exp(-it) + it - 1)

. (3.10)

The covariance matrix of the system noise is computed by simple numerical integration

Qk = x9()Q()x9

T () d=0

t

(3.11)

with the spectral densities

Q(t) =

0 0 0

0 0 00 0 Q3(t)

, (3.12)

where Q3(t) = diag( qan(t) , qae(t) , qah(t) ) are spectral densities of the three acceleration

states.

The models described above are similar to those in Schwarz et al (1989) with one

difference. In Schwarz et al (1989), a carrier phase cycle slip is treated as a random state and

estimated by the augmented Kalman filter algorithm, whereas in our approach, a cycle slip

occurring on a satellite is treated as a constant bias to be recursively estimated and corrected by

a two-stage Kalman filter.

3.2 OBSERVATION MODEL

Observations are used for updating and correcting the Kalman filter states predicted by

the system model. Basically, the GPS system provides two fundamental observables for

geodetic surveying purposes. They are the code pseudo-range observable and the carrier phase

29

observable. For most of the navigation/geodetic GPS receivers, another type of observable, the

instantaneous phase rate (Doppler frequency) is also output for instantaneous velocity

determination of the moving vehicle.

The basic observation equation for raw pseudorange observations, p, is (Lachapelle,

1990):

p = + c(dt - dT) + d + dion + dtrop + (p) (3.13)

where is the geometric range between the receiver antenna and a satellite,

c is the speed of light,

dt, dT are the satellite and receiver clock errors respectively,

d is the range error resulted from satellite orbital errors,

dion , dtrop are the ionospheric and tropospheric corrections, respectively,

and (p) is the pseudo-range measurement noise.

The pseudorange measurements are instantaneous and unambiguous, but their high

measurement noise level limits their use in practice for precise positioning. The typical

measurement noise of pseudoranges is in the order of 1 to 4 metres for C/A code and within

the submetre level for P code. Furthermore, pseudo-range observables are vulnerable to

multipath influences which can often result in errors up to 20 m level in C/A code

measurements (Lachapelle, 1990).

The observation equation for raw carrier phase observations,, is (Wells et al, 1986;

Lachapelle, 1990)

= + c(dt - dT) +N + d - dion + dtrop + () (3.14)

30

where = -measured cycles ( in meters) ,

N is the initial carrier phase cycle ambiguity,

() is the carrier phase measurement noise,

and is the wavelength of the carrier signal (metre).

The main advantages of the carrier phase observable are its low measurement noise

level (usually about 2 mm) and its low sensitivity to multipath effects (usually not exceeding

0.25). But, in order to use carrier phase observables to obtain precise and reliable kinematic

positioning results, the initial carrier phase ambiguity N should be correctly resolved before

kinematic data collection begins. Also, the lock on carrier signals of the satellites should be

maintained during data collection unless the cycle slip problem can be effectively solved by

some methods or external sources. A cycle slip is a discontinuity in the received carrier phase

observations, which results in a change in the integer ambiguity N.

Phase rate or Doppler frequency is the time derivative of the phase. The observation

equation of phase rate can be expressed as:

)(ddd)Tdtd(c tropion ++++= &&&&&&&& . (3.15)

where ( . ) denotes the derivative with respect to time. The linearizd form of (3.15) in the

geodetic coordinate system (, , h) is given by (Lu et al, 1990)

h)hzc

hyb

hxa()zcybxa()zcybxa(

+

+

+

+

+

+

+

+

= &

)(ddd)Tdtd(cVhcosR

VRV

tropionhen ++++

+

+

+ &&&&&& ,

(3.16)

31

where )zz(zx

)yy(yx

)xx(x

a rR2

rR2

rR2

2&&&&&&

+

+

=

)zz(zy

)yy(y

)xx(xy

b rR2

rR2

2rR

2&&&&&&

+

+

=

)zz(z

)yy(yz

)xx(xz

c rR22

rR2

rR2

&&&&&&

++

= .

The above raw observables are severely affected by different error sources, such as

orbital errors, satellite clock and receiver clock errors and atmospheric errors. One effective

way to eliminate or reduce these error terms is to form the differenced observables from the

raw ones. Differencing of simultaneous observations usually cancels the common error terms.

In GPS data processing, the widely used differencing modes are the single difference(SD),

double difference(DD) and triple difference(TD) (Remondi, 1984; Wells et al, 1986). In this

research, the (receiver-satellite) double difference (DD) observables of pseudo-range, carrier

phase and phase rate are used as the measurement updates in the Kalman filter for kinematic

GPS position and velocity estimation. Thus, the observation model reads:

p = + dion + dtrop + (p) (3.17)

= + N + dion + dtrop + () (3.18)

)(dd tropion ++= &&&&& . (3.19)

where is the double difference operator between two stations and two satellites (Wells et

al, 1986).

It is noted that double difference observations cancel out the satellite clock errors and

receiver clock errors, and reduce the orbital and atmospheric errors. Another advantage of the

double difference GPS observable is that the integer nature of cycle ambiguity N in (3.18)

32

can be exploited. Once the integer ambiguity is resolved, it can be held fixed in kinematic

surveys. Unfortunately, if cycle slips occur in carrier phase observations, which is often the

case in real kinematic surveys, the phase measurement update equation (3.18) is no longer

valid but biased by a constant value, which in turn results in erroneous position and velocity

estimates in the filter. Therefore, it is necessary to have some ways to check and correct cycle

slips in the observation model. And this is one of the main problems to be dealt with in the

quality control of precise kinematic GPS surveying.

33

CHAPTER 4

RELIABILITY ANALYSIS IN KINEMATIC

DIFFERENTIAL GPS SURVEYS

Reliability analysis in kinematic GPS surveys includes two aspects, the minimum

detectable bias (MDB) in the filter models and the influences of the undetectable bias on the

filtering results. In this Chapter, the bias influence characteristics on position and velocity

estimation are investigated and the MDB values of some common biases in kinematic GPS

surveys are examined. All these analyses can be done prior to the actual survey campaign,

based on the theoretical formulas given in Chapter 2 and the assumed surveying conditions.

Familiarity with the bias influence behaviors and the MDB values of some common biases as

well as their correlation with the satellite number and satellite geometry is very helpful in the

planning stage of kinematic GPS surveying.

4.1 BIAS INFLUENCES ON POSITION AND VELOCITY ESTIMATION IN

KINEMATIC DIFFERENTIAL GPS SURVEYS

This section investigates the theoretical influences of different kinds of biases on the

position and velocity estimation by using two-stage Kalman filter formulation. In kinematic

GPS surveys, the most common biases in the observation (double difference) model are cycle

34

slips in carrier phases, outliers in pseudoranges and outliers in phase rates. The likely biases

when using the system model are the acceleration errors in the constant velocity model and the

acceleration disturbances in the constant acceleration model. In the following sub-sections, the

bias influence for each kind of bias is developed. The total influence of biases are then the

summation of the influences of each kind of bias, since the bias influences on Kalman filtering

results are linear in nature. These theoretical bias influences, based on the adopted models and

assumed surveying environments, lead to a better understanding of the problems and concerns

in kinematic GPS positioning.

In order to inter-compare the magnitude of the different bias influences on position

states and velocity states, we define a scalar term as follows:

Local Bias-to-Noise Ratio in states (LBNR) is the ratio between the bias xi in a

given Kalman state xi , caused by a bias vector b, and its standard deviation xi , i.e.

LBNR = xi / xi .

In the following computations, we used a part of the trajectory of the semi-kinematic

GPS surveying run carried out in Kananaskis Country on Julian day 222, 1990. This part of

the trajectory consisted of 59 epochs (4 minutes duration) and started at 494132 seconds. Six

satellites were available above an elevation angle of 150 in this part of trajectory. The

dilutions of precision for latitude, longitude and height within the adopted trajectory are given

in Figure 4.1. The height dilution of precision is good but relatively poorer than the latitude

and longitude dilutions of precision. The maximum speed of the vehicle reached 80km/h. The

standard deviations assumed for the pseudorange, carrier phase and phase rate observations

were 4 m, 2 cm and 5 cm s-1 . The output data rate was 4 seconds. All the computations were

done using the program package QUALIKIN described in Chapter 6 and the constant velocity

model was employed as the system model of the Kalman filter for bias influence analysis.

35

4944004943004942004941000

1

2

3LatitudeLongitudeHeight

Epoch Time (GMT second)

Dilu

tion

of P

reci

sion

Fig. 4.1 Dilution of Precision for Part of the Trajectory of Day 222, 1990

(a) Influences of Cycle Slips in Carrier Phase Observations

In double difference (receiver-satellite) carrier phase observations, cycle slips can be

modelled as constant biases following the epoch at which they occur. Figure 4.2 shows one

example of the influence of carrier phase cycle slips on the estimated positions and velocities.

Here we assume that one cycle slip 19.02 cm occurs on SV 11 starting from 494132 seconds.

The pseudorange and phase rate observations as well as the system model are assumed bias-

free. In Figure 4.2, , and h denote the scalar errors (LBNR) in latitude, longitude and

height respectively, caused by the assumed one cycle slip. Vn , Ve and Vh are the scalar

errors (LBNR) in north velocity, east velocity and up velocity, respectively. Figure 4.3 shows

another example where two satellites, instead of one, are assumed to have cycle slips.

36

494400494300494200494100-202468

Vn

Fig. 4.2(a) LBNR for latitude and Vn

seconds

LBN

R1 cycle slip simulated

494400494300494200494100-1

0

1

2Ve

Fig. 4.2(b) LBNR for longitude and Veseconds

LBN

R

494400494300494200494100-6

-4

-2

0 hVh

Fig. 4.2(c) LBNR for height and Vhseconds

LBN

R

Fig. 4.2 Cycle slips influences on positions and velocities. Six satellites (SVs 2,6,9,11,13,18 ) are available simultaneously. SV 11 is assumed to have one cycle slip (19.02cm) starting from 494132 seconds. The data interval is 4 seconds.

37

494400494300494200494100-10

0

10

20Vn

Fig. 4.3(a) LBNR for latitude and Vnseconds

LBN

R 2 cycle slips simulated on two satellites

494400494300494200494100-12-10-8-6-4-20

Ve


LBN

R

494400494300494200494100-20

-10

0

10hVh


LBN

R

Fig. 4.3 Cycle slip influences on positions and velocities. Six satellites (SVs 2,6,9,11,13,18 ) are available simultaneously. Two satellites (SVs 11,18) are assumed to have two cycles slip starting from 494132 seconds. The data interval is 4 seconds.

38

From the above examples, we note that carrier phase cycle slips mostly affect the

estimated positions as opposed to the estimated instantaneous velocities. For one cycle slip

occurring on one satellite, the maximum influence on latitude estimates, as shown in Fig.

4.2(a), reached about 8 times their standard deviations, or about 10 cm in absolute magnitude.

The more satellites affected by cycle slips, the more severe the influences. The total influences

of multiple cycle slips are the summation of the individual cycle slip infulence because of the

linear property of the bias effects on the Kalman filtering results. Figures 4.2 and 4.3 also

show that the cycle slips have a permanent or long-term effect on the estimated positions.

Their influences on the estimated positions are not constant but slowly drifting in time with the

changing satellite geometry. Since the time span in these examples was short (4 minutes), the

drift effect was not very significant. Cannon (1991) has shown one example in which the drift

in height component reached ahout 14 cm for a one hour duration, when one cycle slip was

introduced in the carrier phase observations. Therefore, real-time correction and adaptation of

the cycle slips in double difference observations are important to assure the correctness of the

kinematic positioning results.

(b) Influences of Outliers in Pseudorange Observations

We define an outlier herein as an instantaneous bias that only affects the epoch at

which it occurs. Fig. 4.4 shows the influence of a pseudorange outlier of 20 m on SV 9 at

494132 seconds where six satellites were in view and the vehicle was moving at a speed of 36

km h-1 .

39

494400494300494200494100-0.050

-0.025

0.000

0.025

0.050

Vn


LBN

R

a 20 m outlier in pseudorange simulated

494400494300494200494100-0.050

-0.025

0.000

0.025

0.050Ve


LBN

R

494400494300494200494100-0.050

-0.025

0.000

0.025

0.050 hVh

Fig. 4.4(c) LBNR for height and Vh

seconds

LBN

R

Fig. 4.4 Pseudorange outlier influences on positions and velocities. Six satellites (SVs 2,6,9,11,13,18) are in view simultaneously. A single outlier of 20 m is assumed in the pseudorange of SV 11 at 494132 seconds. The data interval is 4 seconds.

40

From the single outlier case shown in Fig. 4.4 and other test computations, we notice

that the magnitude (LBNR) of the influences of pseudorange outliers on positions and

velocities is much smaller than that due to cycle slips and practically can be neglected. This is

caused by the lower a priori standard deviation of pseudorange observations. The pseudorange

outliers mainly affect the positions as opposed to the velocities. Furthermore, such effects are

only limited to the current and subsequent few epochs when no bias occurs on the phase and

phase rate measurements. This means that the Kalman filter can reduce the outlier influence

on the estimated states after a few epochs of measurement updates. Here we assume that the

system model is correct.

(c) Influences of Outliers in Phase Rate Observations

Unlike the carrier phase observation which is the accumulated cycles of the receiveed

carrier signal, phase rate (Doppler) is an instantaneous observation that reflects the changing

range rate from the receiver to the observed satellite. The likely biases in phase rate

observations are outliers present on each single measurement. Fig. 4.5 shows the influences on

the estimated positions and velocities for an outlier of 0.5m s-1 occurring at time 494132 in

the phase rate observation of SV 11.

494400494300494200494100-10123456

Vn


LBN

R

a 0.5 m s outlier simulated -1

41

494400494300494200494100-1

0

1

2Ve


LBN

R

494400494300494200494100-4-3-2-101

hVh


LBN

R

Fig. 4.5 Phase rate outlier influences on positions and velocities. Six satellites (SVs 2,6,9,11,13,18) are available simultaneously. A single outlier of 0.5m s-1 in the phase rate occurs on SV 11 at 494132 seconds. The data interval is 4 seconds.

We note two points from Figure 4.5. Firstly, the outlier in phase rate observations

mainly affects the estimated velocities. This is different from the cycle slip influences which

mainly affect the estimated positions. Secondly, the phase rate outlier influences on velocities

are limited to the epoch at which the outlier occurs and the following few epochs. The Kalman

filter can effectively reduce the outlier influences on the filtering states after a few epochs of

observation updates. This is due to the nature of an outlier, which, by its definition, only

affects one observation at a single epoch.

42

(d) Influences of the Bias in the System Kinematic Model

Besides the biases in the observation model, biases in the system model may exist due

to deviations between the assumed and the actual vehicle motion. According to the Kalman

filter algorithm, system model biases mainly affect the prediction or time propagation of the

filter state vector xk(-) , which in turn influences the computation of the innovation sequence

or predicted residuals. If the measurements are correct (bias-free), then the inaccurately

predicted states xk(-) can be corrected by the measurement updates through the Kalman gain

matrix Kk .

Fig. 4.6 shows one example of the system bias influences on the predicted states xk(-),

i.e. the predicted positions and velocities. Here we assume that within a constant velocity

model (3.1), a single acceleration bias of 2 m s-2 occurs on the system state Vn , the main

direction of the trajectory, for 6 consecutive epochs from time 494132 to 494152. The data

rate is 4 seconds. The local bias-to-noise ratio (LBNR) in this example is defined by

(-)/(-) .

494400494300494200494100-2

0

24

6

8

10

h

Fig. 4.6(a) LBNR for Latitude, Longitude and Height


LBN

R

a system bias simulated in six epochs

43

494400494300494200494100-202468

10

Vn

Ve

Vh

Fig. 4.6(b) LBNR for Northing, Easting and Up Velocities


LBN

R

Fig. 4.6 System bias influences on predicted positions and velocities. An acceleration bias of 2 m s-2 occurs on the system state Vn at 6 consecutive epochs from time 494132

to 494152. Six satellites (SVs 2,6,9,11,13,18) are available simultaneously. The data interval is 4 seconds.

From Fig. 4.6, one can see that the acceleration bias in the system model severely

affects the corresponding predicted states k(-) and Vn(-) . The absolute magnitudes of

influences reach 16 metres in latitude and 8 m s-1 in northing velocity. These influences are

limited to the epochs at which the system bias occurs. The shorter the update or data rate

interval, the smaller the magnitude of influence. For instance, under the same conditions but at

an update interval of 1 second, the magnitudes of influence reduce to 1 metre in latitude and 1

m s-1 in northing velocity. In order to limit the system bias influences on the innovation

sequence and facilitate the detection of observation biases, a high data update rate (e.g. 1

second or higher), is recommended.

44

4.2 MDB IN KINEMATIC DIFFERENTIAL GPS SURVEYING MODEL

Minimum Detectable Bias (MDB) analysis is an important tool for Kalman filter

design. It tells us about the filter's theoretical ability to detect the concerned biases under the

assumed surveying conditions. Based on the formulas given in Section 2.3.1 of Chapter 2, the

MDB values of some typical biases in the kinematic GPS position and velocity estimation

models are investigated in this section. These biases include the carrier phase cycle slips,

outliers in pseudoranges and outliers in phase rate measurements.

In all the following computations, the same trajectory from Julian Day 222, 1990

described in Section 4.1 was used. The non-centrality parameter for MDB computations is set

to o = (4.13)2 , which corresponds to 0.1% and 20% probabilities of Type I and Type II

errors, respectively, in a one-dimensional hypothesis test. Corresponding to the testing

statistics and the testing procedure used in the bias detection and identification in kinematic

GPS surveying, the local MDB (or the instantaneous MDB at each epoch) for a single bias

case is of great importance, which, from equation (2.50), is given by

b0 i =

0 Si

T Qvi-1 Si ,

(4.1)

where i is the index for epochs. Here we define the single bias case as the case where no

biases other than the specified one are presented in the processing models. The relationship

between the local MDB values and the satellite geometry is also examined.

45

(a) MDB for Carrier Phase Cycle Slips

Cycle slips are common biases occurring during carrier phase measurements. They can

be treated as constant biases in the receiver-satellite double difference observables if the initial

carrier phase ambiguities are resolved and held fixed during the kinematic survey.

Fig. 4.7 shows the improvements in minimum detectable bias values at each epoch in

double difference carrier phase observations of SV 02 with different satellite coverages. Fig.

4.8 shows the corresponding GDOP values of different satellite coverages. The four, five and

six satellite coverages used for comparisons are (SVs 6, 2, 9, 11), (SVs 6, 2, 9, 11, 13) and

(SVs 6, 2, 9, 11, 13, 18). Here we assume that only the concerned biases, i.e. cycle slips, are

presented in the observations and the system model is errorless.

4944004943004942004941000.00.20.40.60.81.01.2

4 sat. coverage5 sat. coverage6 sat. coverage

seconds

Loca

l MD

B (m

)

Fig. 4.7 Local MDB Improvement in Double Difference Carrier Phase Observations on SV

02 with the Different Satellite Coverages.

46

4944004943004942004941002

4

6

8

10

124 sat. coverage5 sat. coverage6 sat. coverage

Seconds

GD

OP

Fig. 4.8 GDOP improvements with the different satellite coverages

Figures 4.7 and 4.8 show that the MDB values at each epoch in the double difference

carrier phase observations of SV 02 with the 4 satellite coverage, where the GDOP values

reach approximately 10, are about 1 metre (i.e. 5 cycles of L1 frequency). With the 5 and 6

satellite coverages, where the GDOP values are around 3.2, the corresponding MDB values

drop to about 0.1 metre (i.e. 0.5 cycle of L1 frequency). These MDB values reflect the best or

theoretical values that the statistical testing can detect with a given testing power for a single

bias present in

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